The Laplace series, so named the generalized Fourier series in spherical harmonics on the unit sphere, has
been extensively studied. One essential ingredient for understanding these series is the addition formula for
spherical harmonics, which states that the reproducing kernel of the orthogonal projection operator from \(L^2({\mathbb {S}^{d-1}})\) 
onto the space of spherical harmonics of degree \(n\)  can be written as \(Z_n ({\langle }\xi ,\eta {\rangle })\) , where \(Z_n\) 
is a Gegenbauer polynomial of degree \(n\)  in one variable and \({\langle }\xi ,\eta {\rangle }\)  is the Euclidean inner product
of \(\xi ,\eta \in {\mathbb {S}^{d-1}}\) . Because of this closed formula of the reproducing kernels, much of the study of the
Laplace series can be reduced to the study of the Fourier-Gegenbauer series of one variable [1]}, [2]}, [3]}.
